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Characterization of Quantum Entanglement Via a Hypercube of Segre Embeddings

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Characterization of Quantum Entanglement Via a Hypercube of Segre Embeddings Characterization of quantum entanglement via a hypercube of Segre embeddings Joana Cirici∗ Departament de Matemàtiques i Informàtica Universitat de Barcelona Gran Via 585, 08007 Barcelona Jordi Salvadó† and Josep Taron‡ Departament de Fisíca Quàntica i Astrofísica and Institut de Ciències del Cosmos Universitat de Barcelona Martí i Franquès 1, 08028 Barcelona A particularly simple description of separability of quantum states arises naturally in the setting of complex algebraic geometry, via the Segre embedding. This is a map describing how to take products of projective Hilbert spaces. In this paper, we show that for pure states of n particles, the corresponding Segre embedding may be described by means of a directed hypercube of dimension (n − 1), where all edges are bipartite-type Segre maps. Moreover, we describe the image of the original Segre map via the intersections of images of the (n − 1) edges whose target is the last vertex of the hypercube. This purely algebraic result is then transferred to physics. For each of the last edges of the Segre hypercube, we introduce an observable which measures geometric separability and is related to the trace of the squared reduced density matrix. As a consequence, the hypercube approach gives a novel viewpoint on measuring entanglement, naturally relating bipartitions with q-partitions for any q ≥ 1. We test our observables against well-known states, showing that these provide well-behaved and fine measures of entanglement. arXiv:2008.09583v1 [quant-ph] 21 Aug 2020 ∗ [email protected] † [email protected] ‡ [email protected] 2 I. INTRODUCTION Quantum entanglement is at the heart of quantum physics, with crucial roles in quantum information theory, superdense coding and quantum teleportation among others. An important problem in entanglement theory is to obtain separability criteria. While there is a clear definition of separability, in general it is difficult to determine whether a given state is entangled or separable. A refinement of this problem is to quantify entanglement on a given entangled state. This is a broadly open problem, in the sense that there is not a unique established way of measuring entanglement, which might depend on the initial set-up and applications on has in mind. There is, however, a general consensus on the desirable properties of a good entanglement measure [1, 2]. Directly attached to measuring entanglement is the notion of maximally entangled state, central in teleportation protocols. Many different entanglement measures and the corresponding notions of maximal en- tanglement have been proposed. Methods range from using Bell inequalities, looking at inequalities in the larger scheme of entanglement witnesses, spin squeezing inequalities, en- tropic inequalities, the measurement of nonlinear properties of the quantum state or the approximation of positive maps. We refer to the exhaustive reviews [2–4] for the basic aspects of entanglement including its history, characterization, measurement, classification and applications. A particularly simple description of separability of quantum states arises naturally in the setting of complex algebraic geometry. In this setting, pure multiparticle states are identified with points in the complex projective space PN , the set of lines of the complex space CN+1 that go through the origin. Entanglement is then understood via the categorical product of projective spaces: the Segre embedding. This is a map of complex algebraic varieties 1 (n) 1 2n 1 P P P − , (1) × ···× −→ whose image, called the Segre variety, is described in terms of a family of homogeneous quadratic polynomial equations in 2n variables, where n is the number of particles. The points of the Segre variety correspond precisely to separable states (see for instance [5, 6]). In this paper, we exploit this geometric viewpoint to show that the image of (1) is in fact given by the intersection of all Segre varieties defined via bipartite-type Segre maps 2ℓ 1 2n−ℓ 1 2n 1 P − P − P − , for all 1 ℓ n 1. (2) × −→ ≤ ≤ − Specifically, we show that a state is q-partite if and only if it lies in q of the images of Segre maps of the form (2). We do this after showing that the Segre embedding (1) may be decomposed in various equivalent ways, leading to a hypercube of dimension (n 1) whose edges are bipartite-type Segre maps. In this framework all the information about− the separability of the state in contained in the last applications (2) of the hypercube. There are numerous approaches to entanglement via Segre varieties that are related to the present work [7–13]. The hypercube viewpoint presented here is a novel approach that connects the notions of bipartite and q-partite in a geometric way. While the above results are extremely precise and intuitive, they are purely algebraic. In order to build a bridge from geometry to physics, we introduce a family of observables n,ℓ , with 1 ℓ n 1, which allow us to detect when a given n-particle state belongs to{J each} of the≤ images≤ of− the bipartite-type Segre maps (2). As a consequence, we obtain that a state is q-partite if and only if at least q of the observables n,ℓ vanish on this state, completely identifying the sub-partitions of the system. Each ofJ these observables are related to the trace of the squared reduced density matrix, also known as the bipartite non-extensive Tsallis entropy with entropic index two [14, 15]. They are always positive on 3 entangled states and our first applications indicate that they provide well-behaved and fine measures of entanglement. We briefly explain the contents of this paper. We begin with a warm-up Section II where we detail the theory and results for the well-understood settings of two- and three-particle states. In Section III we develop the geometric aspects of the paper. In particular, we describe the Segre hypercube and prove Theorem III.6 on geometric decomposability. The main result of Section IV is Theorem IV.3, where we match geometric decomposability with our family of observables related to non-extensive entropic measures. The two theorems are combined in Section V, where we study entanglement measures for pure multiparticle states 1 of spin- 2 and apply our observables on various well-known multiparticle states. ACKNOWLEDGMENTS We would like to thank Joan Carles Naranjo for his ideas in the proof of Lemma A.1. J. Cirici would like to acknowledge partial support from the AEI/FEDER, UE (MTM2016- 76453-C2-2-P) and the Serra Húnter Program. J. Salvadó and J. Taron are partially supported by the Spanish grants FPA2016-76005-C2-1-PEU, PID2019-105614GB-C21, PID2019-108122GB-C32, by the Maria de Maeztu grant MDM-2014-0367 of ICCUB, and by the European INT projects FP10ITN ELUSIVES (H2020-MSCA-ITN-2015-674896) and INVISIBLES-PLUS (H2020-MSCA-RISE-2015-690575). II. WARM-UP: TWO AND THREE PARTICLE STATES We begin with the well-understood example of two particle entanglement. The initial set-up consists in two particles which can be shared between two different observers, A and B, that can perform quantum measures to each of the particles. A general pure state for 1 two spin- 2 particles can be written as ψ = z 00 + z 01 + z 10 + z 11 , (3) | iAB 0 | i 1 | i 2 | i 3 | i 2 where zi are complex numbers that satisfy the normalization condition zi = 1. The labels AB, which will be often omitted, indicate that A is acting on the first| particle| while B is acting on the second, so P ij := i j , for i, j 0, 1 . | i | iA ⊗ | iB ∈{ } Entanglement is a property that can be inferred from statistical properties of different measurements by the observers of the system. In particular, we can compute the sum of the expected values of A measuring the spin of the state ψ in the different directions. With this idea in mind, we define: | i 3 2 A B(ψ) := 2 ψ σi I2 ψ , J ⊗ − | h | ⊗ | i | i=0 ! X where σi, for i =1, 2, 3, denote the Pauli matrices, σ0 = I2 is the identity matrix of size two and denotes the Kronecker product. Using the expression (3) for ψ we obtain ⊗ | i 2 A B(ψ)=4 z0z3 z1z2 . J ⊗ | − | 4 It is well-known that the state ψ is entangled if and only if | i z z z z =0. 0 3 − 1 2 Therefore we find that ψ is a product state if and only if A B (ψ)=0. When A B (ψ) > 0 we have an entangled| i state, which is considered to beJ maximal⊗ ly entangledJ when⊗ the observable reaches its maximum value at A B(ψ)=1. J ⊗ The measure given by A B(ψ) may also be interpreted in terms of the density matrix operator. Indeed, the densityJ ⊗ matrix ρ for a pure state ψ is given by | i z0z0 z0z1 z0z2 z0z3 z z z z z z z z ρ = ψ ψ = 1 0 1 1 1 2 1 3 . | i h | z2z0 z2z1 z2z2 z2z3 z3z0 z3z1 z3z2 z3z3 Compute the reduced matrix for the subsystem A by means of the partial trace on B, z0z0 + z1z1 z0z2 + z1z3 ρA = TrBρ = . z2z0 + z3z1 z2z2 + z3z3 The trace in the subsystem A of the square reduced matrix gives Trρ2 =1 2 z z z z 2. A − | 0 3 − 1 2| This magnitude is refereed in the literature as the Tsallis entropy or q-entropy with q = 2 and is directly related with the proposed measure by, 2 A B(ψ)=2 1 TrAρA . J ⊗ − Recall that by the Schmidt Decomposition, and after choosing a convenient basis, any pure 2-particle state may be written as ψ = cos(θ) 00 + sin(θ) 11 | i | i | i where θ [0,π/4] is the Schmidt angle, which is known to quantify entanglement (see for instance∈ [16]).
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